Analysis of Pressure-Falloff Tests Following Cold-Water Injection
- R.B. Bratvold (Stanford U.) | R.N. Horne (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Formation Evaluation
- Publication Date
- September 1990
- Document Type
- Journal Paper
- 293 - 302
- 1990. Society of Petroleum Engineers
- 5.5.1 Simulator Development, 5.2.1 Phase Behavior and PVT Measurements, 5.4.1 Waterflooding, 5.2 Reservoir Fluid Dynamics, 6.5.2 Water use, produced water discharge and disposal, 5.6.1 Open hole/cased hole log analysis, 5.3.4 Reduction of Residual Oil Saturation, 4.1.2 Separation and Treating, 5.1.5 Geologic Modeling, 1.8 Formation Damage, 5.6.4 Drillstem/Well Testing, 5.5 Reservoir Simulation, 4.1.5 Processing Equipment, 5.6.5 Tracers
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Summary. This paper presents generalized procedures to interpret pressure injection and falloff data following cold-water injection into a hot-oil reservoir. The relative permeability characteristics of the porous medium are accounted for, as is the temperature dependence of the fluid mobilities. It is shown that the saturation and temperature gradients have significant effects on the pressure data for both the injection and falloff periods. The matching of field data to type curves generated from periods. The matching of field data to type curves generated from analytical solutions provides estimates of the temperature-dependent mobilities of the flooded and uninvaded regions. The solutions also may be used to provide estimates of the size of the invaded region, the distance to the temperature discontinuity, heat capacities, and wellbore-storage and skin effects.
Numerous full-field waterflooding projects are currently under way throughout the world to improve recovery. In many large oilfields, water injection is initiated during the early stages of reservoir development. Exploratory wells are tested for injectivity, and injectors are tested during field operation. If properly interpreted, these tests can give information about the progress of the flood (i.e., frontal advance), residual oil saturation, the flow characteristics of the virgin formation, and near-wellbore damage.
In a water-injection well test, the injected fluid usually has a temperature different from the initial reservoir temperature. During injection, both a saturation and a temperature front propagate into the reservoir. Furthermore, because of differences in oil and water properties, a saturation gradient is established in the reservoir. The properties, a saturation gradient is established in the reservoir. The water saturation is highest close to the well and continuously decreases with distance from the well. Ahead of this invaded region is the unflooded oil bank at initial water saturation.
For the interpretation of well-test data, the most important temperature-dependent fluid property is the viscosity. The viscosity of both oil and water may change by an order of magnitude between 50 and 572 degrees F, with the major change occuring between 50 and 212 degrees F. This temperature effect strongly influences the fluid mobilities, and hence the saturation gradient and the transient pressure response. The total fluid mobility changes continuously in the invaded region and has to be accounted for in reservoir modeling and data interpretation.
Many different models have been introduced for the analysis of water-injection and falloff tests. Typically, these models neglect the temperature effects, the saturation gradient, or both. Refs. 2 and 3 provide reasonably complete reviews of previous works.
For this paper, the most important reference is Fayers' extension of the fractional flow theory of Buckley and Leverett to account for a radial temperature gradient in the reservoir. Fayers' work was put into a mathematical framework by Karakas et al. and Hovdan. Hovdan also used this incompressible-fluids solution to derive a pressure-transient solution for the late stages of a cold-water-injection test.
Recently, Abbaszadeh and Kamal presented procedures to analyze falloff data from water-injection wells. Their procedures are based on analytical solutions not presented in their original paper and include the effect of the saturation gradient in the invaded region. Nonisothermal effects were not considered.
In summary, a number of studies pertaining to well-test analysis of injection and falloff tests have been presented. However, none of these account for both of the two most important effects in a typical waterflood: the saturation gradient and the temperature effect.
The principal objectives of this paper are (1) to derive analytical solutions that include the most important effects in a nonisothermal water-injection/falloff test, (2) to examine the parameters that influence the well injectivity, and (3) to present procedures to obtain detailed and accurate information about the important reservoir and fluid properties in a waterflood. Specifically, we consider the pressure behavior at the well resulting from the simultaneous flow of pressure behavior at the well resulting from the simultaneous flow of oil and water in a reservoir with a radial temperature gradient. Analytical solutions that account for the effects of temperature and saturation gradients are derived and discussed. Consequences of neglecting the temperature and saturation effects are illustrated.
Solutions for linear systems, including the effects of linear boundaries in cylindrical reservoirs, were presented by Bratvold and Larsen.
Fig. 1 presents a schematic of the reservoir configuration considered. The reservoir is assumed to be cylindrical with the well at the center. The well penetrates the entire formation thickness, and fluid is injected at a constant rate. The reservoir is assumed to be a uniform, homogeneous porous medium, completely saturated with oil and water. Liquid compressibilities are assumed to be constant, while the viscosities are assumed to be functions of temperature only. Neglecting effects of gravity, as well as heat transfer to the surrounding formation, permits the use of a ID radial model.
Injection Period. The transient, nonisothermal two-phase flow of oil and water requires that saturations, pressures, and temperatures be determined simultaneously at any time. Furthermore, because cold-water injection into a hot-oil reservoir is a moving-boundary problem, it cannot be solved with standard linear techniques, such as problem, it cannot be solved with standard linear techniques, such as eigenfunction expansion, integral transforms, or Green's function methods.
To circumvent the problem of simultaneously solving the coupled second-order conservation equations, we derive an alternative approximate solution to the injection problem using a two-step procedure. procedure. Step 1. Assume incompressible fluids. Then use fractional flow theory to solve the resulting first-order coupled energy- and mass-conservation equations. This essentially amounts to decoupling the equations for saturation and temperature from the pressure equation. The saturation profile obtained is a Buckley-Leverett profile including (convective) temperature effects. profile including (convective) temperature effects. Step 2. With the saturation and temperature profiles and the mobilities and diffusivities known from Step 1, solve the diffusion equation for pressure by assuming that the fluid compressibilities are small and constant. Hence, the pressure distribution in the system is obtained by superimposing pressure-transient effects on a saturation profile known a priori.
Fig. 2 shows an example of a saturation and temperature distributions as functions of the similarity transform , as calculated from the Buckley-Leverett model and including temperature effects. Note that the profile exhibits two saturation discontinuities. In addition to the discontinuity depicted by the standard Buckley-Leverett theory, the saturation distribution shows a second discontinuity caused by the step-change in temperature. The magnitude of the saturation change at the temperature discontinuity is related to the ratio between the mobility ratios in the hot and cold zones. The saturation distribution obtained from a numerical simulator is superimposed on the analytical saturation profile. The simulations were performed with a two-phase, 2D, black-oil simulator developed by Nyhus that is described later in the paper.
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